航天器工程：航天器建模课件

The Many-Body problem
• The other approach to the solution is the numerical integration of the differential equations of motion. This can be done using high speed computers, and is called “special perturbations”. • Perturbations may be divided in two further classes: Periodic and Secular. • Any disturbance of the reference orbit that is repeated with a given period of revolution is termed a periodic perturbation.
The Many-Body problem
• Since the orbits of the planets in the solar system are very accurately elliptical, the solar system can be first approximated neglecting the planet-planet gravitational attraction: in this way the problem is reduced to a set of two-body problems, one for each planet. • A better approximation is obtained assuming that the effect of planet-planet gravitational attraction is a perturbation of the elements of the elliptical orbits. • Analytical expressions, valid for a period of time, have been found in this way. These are called “general perturbations”. • Effects like the advance of perihelium or the retroregression of the ascending node of a planetary orbit have been found in this way.
The Many-Body problem
• Given at any time the positions and velocities of three or more massive particles, moving under their mutual gravitational forces, the masses also being known, calculate their positions and velocities at any other time. • This problem is orders of magnitude more difficult than the two-body problem we have considered. • If the bodies are not point-like and not spherical, additioBiblioteka Baidual complexity is added. • No general analytical solution has been found in the last three centuries, despite of the enormous amound of work carried out by people like Euler, Lagrange, Poincare, etc.
The Many-Body problem
• There are 10 known integrals of the motion, which were already known to Euler: since then, no further integrals have been discovered. • Solution in particular cases has been found by Lagrange, which are of interest in astrodynamics and in astronomy. • Families of periodic orbits of a small test particle in the field of two masses orbiting in undisturbed circular orbits have been found at the epoch of Poincare’. These are of interest as an approximation of a space vehicle in Earth-Moon space.
• A secular perturbation, instead, causes a change proportional to time. Typical example: the advance of perihelion. • In the case of Mercury, the advance (greatly exaggerated in the figure) is 43 arcsec/year. • Only part of it is explained by newtonian dynamics. General relativity is needed to fully explain it. • In the case of the binary pulsar PSR1913+16 (the Hulse-Taylor pulsar, used to demonstrate the existence of gravitational waves) the advance is as large as 4.2o/year